Theorem redcwlpo 16832

Description: Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 16831). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10603 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
redcwlpo	|- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Distinct variable group:   x, y

Proof of Theorem redcwlpo

Dummy variables f i j z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> A. x e. RR A. y e. RR DECID x = y )
2		elmapi 6903	. . . . . . . . 9 |- ( f e. ( { 0 , 1 } ^m NN ) -> f : NN --> { 0 , 1 } )
3	2	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> f : NN --> { 0 , 1 } )
4		oveq2 6057	. . . . . . . . . . 11 |- ( i = j -> ( 2 ^ i ) = ( 2 ^ j ) )
5	4	oveq2d 6065	. . . . . . . . . 10 |- ( i = j -> ( 1 / ( 2 ^ i ) ) = ( 1 / ( 2 ^ j ) ) )
6		fveq2 5669	. . . . . . . . . 10 |- ( i = j -> ( f ` i ) = ( f ` j ) )
7	5, 6	oveq12d 6067	. . . . . . . . 9 |- ( i = j -> ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) ) )
8	7	cbvsumv 12042	. . . . . . . 8 |- sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = sum_ j e. NN ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) )
9	3, 8	trilpolemcl 16813	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR )
10		1red 8288	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> 1 e. RR )
11		eqeq1 2239	. . . . . . . . 9 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> ( x = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
12	11	dcbid 846	. . . . . . . 8 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> (DECID x = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
13		eqeq2 2242	. . . . . . . . 9 |- ( y = 1 -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
14	13	dcbid 846	. . . . . . . 8 |- ( y = 1 -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
15	12, 14	rspc2v 2933	. . . . . . 7 |- ( ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR /\ 1 e. RR ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
16	9, 10, 15	syl2anc 411	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
17	1, 16	mpd 13	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 )
18	3, 8	redcwlpolemeq1 16831	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> A. z e. NN ( f ` z ) = 1 ) )
19	18	dcbid 846	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> DECID A. z e. NN ( f ` z ) = 1 ) )
20	17, 19	mpbid 147	. . . 4 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID A. z e. NN ( f ` z ) = 1 )
21	20	ralrimiva 2615	. . 3 |- ( A. x e. RR A. y e. RR DECID x = y -> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
22		nnex 9242	. . . 4 |- NN e. _V
23		iswomninn 16827	. . . 4 |- ( NN e. _V -> ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 ) )
24	22, 23	ax-mp 5	. . 3 |- ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
25	21, 24	sylibr 134	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> NN e. WOmni )
26		nnenom 10795	. . 3 |- NN ~~ om
27		enwomni 7460	. . 3 |- ( NN ~~ om -> ( NN e. WOmni <-> om e. WOmni ) )
28	26, 27	ax-mp 5	. 2 |- ( NN e. WOmni <-> om e. WOmni )
29	25, 28	sylib 122	1 |- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2812   {cpr 3689   class class class wbr 4108   omcom 4711   -->wf 5347   ` cfv 5351  (class class class)co 6049   ^m cmap 6881   ~~ cen 6972  WOmnicwomni 7453   RRcr 8125   0cc0 8126   1c1 8127   x. cmul 8131   / cdiv 8945   NNcn 9236   2c2 9287   ^cexp 10899   sum_csu 12034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-womni 7454  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-ico 10226  df-fz 10342  df-fzo 10476  df-seqfrec 10809  df-exp 10900  df-ihash 11137  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-clim 11960  df-sumdc 12035

This theorem is referenced by: (None)